Now we explore a scenario in which a properly selected single dielectric inclusion
within a host medium whose permittivity (or effective permittivity) is near zero may lead
to simultaneously having effective permittivity and permeability near zero. It was shown
in previous work 25 that with an ENZ host medium one can also achieve an effective
MNZ by periodically loading this host medium with inclusions of proper dimensions and
permittivity. In this section we show that this can be extended into a non-periodic case
with arbitrary shape of the cross section, and that even within one unit cell, which can be
arbitrarily large and not limited to a sub-wavelength size, we can still have both effective
permittivity and permeability near zero.
The first proposed structure in which we theoretically demonstrate the EMNZ
behavior is shown in Fig. 3, where we have an input channel that is a 2D air-filled
parallel plate waveguide (region 1) feeding the 2D region of interest (region 2) which is
filled with a host medium whose permittivity follows the Drude model
(
)
2 0 1 p h i ω ε ε ω ω = − + Γ , where ωp is the plasma frequency, Γ is the collision frequency
(that is assumed to be zero, i.e., a lossless medium). This medium obviously behaves as
ENZ at the plasma frequency. This region is enclosed by PEC walls and is connected to
18
the inclusion to be a 2D dielectric rod of radius R and permittivity εi , which when embedded in an ENZ host, one can achieve an effective permeability shown in 25 to be
2 , 1 0 (k R) 2 1 ( ) (k R) h cell i eff o cell cell i i A R J A A k R J π µ = µ + at ωp (5)
where Acell is the total area of the unit cell and is equal to a a× , Ah cell, =Acell−πR2, and ki = ω ε µi o . As mentioned before, we are interested in a zero effective
permeability, thus we search for (k Ri ) versus the normalized radius R
a for which we
achieve zero effective permeability are shown in Fig. 4(a). Since our main goal is to
achieve a large enough inclusion-free region where the EMNZ behavior is exhibited, we
choose to investigate a scenario where R
a is very small, e.g., 115for three different cases
namely
a=λ
, 1.5λ
, and 3λ
. The required dielectric constant for the rod inclusion to achieve zero effective permeability turns out to be 33.23, 14.77, and 3.69 respectively. InFig. 5, for the three cases, it is verified that even within a single unit cell that can be
several wavelengths large we achieve the EMNZ behavior of “opening up” the space, as
suggested by the perfect transmission between the input and output ports and the uniform
phase within the proposed EMNZ region. As shown in Fig. 6, the EMNZ behavior shows
to be preserved regardless of the location of the single dielectric rod within the unit cell,
which provides us with a large enough “real-state” with effective epsilon and mu near
zero, within which we can exploit the exotic EMNZ features discussed in previous
19
Moreover, we show that for a fixed 2D cross-sectional area, radius and permittivity of
the dielectric inclusion, we would always get the structure to behave as EMNZ regardless
of its arbitrary cross-sectional shape. Therefore, using the same findings of the previous
discussion, for the case of a normalized radius 115 and a unit cell area of
λ×λ
we investigate two different cross-sectional shapes for the unit cell to prove our concept. InFig. 7 we use a circular-cross-section enclosure, while in Fig. 8 we have an arbitrarily-
shaped-cross-section enclosure, and we note that the EMNZ behavior is still preserved
for both cases. As shown from the required permittivities for achieving the near-zero
effective permeability, it is clear that as we increase the size of the region of interest,
within which we are aiming for EMNZ behavior, we need smaller values of
permittivities, for example the value required for
3λ×3λ
region is 3.69 which in the range of permittivities of readily available materials like Lithium Niobate. Fig. 4(b)shows the required dielectric constant of the rod to achieve a zero effective permiability
for
8
R= λ versus the normalized area Acell2
λ . The numerical simulation has been
20
Fig. 3 Geometry of the proposed 2D structure in which a dielectric rod is inserted in an
ENZ host medium, causing region 2 to behave as EMNZ medium.
Fig. 4 Investigating the dependence of the dielectric constant of the rod to achieve
EMNZ behavior on geometrical aspects, Plot of the (a) required normalized index of
refraction of the rods as a function of the normalized radius R/a, in order , (b) required permittivy of the rods as a function of the normalized area to achieve µeff =0.
21
Fig. 5 The proposed 2D structure to exhibit EMNZ behavior for a normalized radius
/
R a of
1/ 15
, (a) Snapshot of the magnetic field distribution fora= λ
,a= λ1.5
,3
a= λ
, (b) Amplitude and (c) phase of the magnetic field distribution fora= λ
,a= λ1.5
22
Fig. 6 Similar to Fig. 5, except the 2D dielectric inclusion is moved to the corner of the
23
Fig. 7 The proposed 2D structure to exhibit EMNZ with a circular cross section
behavior for a normalized radius R a of /
1/ 15
(a) Snapshot of the magnetic field distribution, Phase of (b) magnetic field Hz, (c) x-component of the electric field, and (d) y-component of the electric field over the EMNZ region.24
Fig. 8 Similar to Fig. 7, except the cross section of the structure is chosen to be
arbitrary.